The functor \(A^{\min}\) on \(p\)-local spaces
type: Item
instance of: scholarly article
title: The functor \(A^{\min}\) on \(p\)-local spaces
zbMATH Open document ID: 1107.55005
author: Paul Selick
author: Jie Wu
published in: Mathematische Zeitschrift
publication date: 2006-11-09T00:00:00Z
review text: It is know that there is a natural isomorphism \(H_*(\Omega \Sigma X,\mathbb F)\cong T(\tilde{H}_*(X,\mathbb F))\) for a field \(\mathbb F\) and that the tensor coalgebra \(T(V)\) has a further decomposition by using the Poincaré-Birkhoff-Witt Theorem, and one may expect that there exists a functorial non-trivial homotopy decomposition of \(\Omega \Sigma X\). The authors gave the finest such functorial decomposition for a \(p\)-torsion suspension \(X\) in their previous paper [On natural coalgebra decompositions of tensor algebras and loop suspensions. Mem. Am. Math. Soc. 701, 109 p. (2000; Zbl 0964.55012)]. In this paper, they generalize the previous results to loop suspensions of an arbitrary \(p\)-local path connected space \(X\). In particular, they show that there is a functorial decomposition \[ \displaystyle \Omega \Sigma X\simeq A^{\text{ min}}(X)\times \Big( \bigvee_{n=2}^{\infty}Q_n^{\text{ max}}(X)\Big) \] if \(X\) is a path connected \(p\)-local CW complex with several other useful results. To prove these results, they define a specific map \(\pi_X:\Big( \bigvee_{n=2}^{\infty}Q_n^{\text{ max}}(X)\Big)\to\Sigma X\) and show that \(\Omega \pi_X\) has a left homotopy inverse by using the calculations of algebraic James-Hopf maps. The key of the generalization is the fact in modular representation theory any idempotent in \((\mathbb Z/p)(\Sigma_n)\) lifts to an idempotent in \((\mathbb Z_{(p)})(\Sigma_n)\), where \(\Sigma_n\) stands for the symmetric group of \(n\)-letters.
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 55P40
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 5071437
zbMATH Keywords: \(p\)-local space
zbMATH Keywords: product decomposition
zbMATH Keywords: loop space
zbMATH Keywords: suspension
zbMATH Keywords: tensor product
zbMATH Keywords: Hopf algebra
zbMATH Keywords: James-Hopf map
zbMATH Keywords: James construction
MaRDI profile type: Publication
full work available at URL: s00209-005-0909-3
OpenAlex ID: W2008815328
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DOI: 10.1007/S00209-005-0909-3
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description: scientific article; zbMATH DE number 5071437